Most of the miso soups I've ever seen sold, particularly outside of Japan, are made with white miso. I understand this is the most popular kind for soup.

White miso is made differently to red miso (higher rice to -typically- soybean content?) and has a less salty, sweeter, 'milder' taste. From my experience, it tends to have less 'bits' in it too once dissolved.

Red miso is more, what I regard as, 'country style'. The taste is stronger, and it is usually noticeably saltier.

I've only been able to purchase artisan grade Japanese miso in the UK quite recently (earlier this year) and so far only in the red style. This has a rich, developed taste and makes a hearty, delicious soup (it's amazing in Korean style soups too). It would be unfair of me to make a direct comparison of this 'primo' miso with the white stuff you get in London stores.

If you are going to spend time on making a superb stock, I wouldn't use the red miso though. That hearty taste will probably overpower it.

My view is that a good dashi stock and a good white miso will meld together in a blissful partnership.

A soup based on red miso is usually dominated by the taste of miso.

As my dashi making skills are only so-so (I only make it occasionally, the rest of the time I rely on the best hon-dashi pre-pared dashi I can get) a good red miso makes all the difference. So much so, that now I can source and replenish good quality miso the 'miso soup' I now make most doesn't need any dashi at all (recipe came courtesy of another FTer - QShoeGuy^)

---
There are other misos apart from red and white, but these two are the most common. Miso can also often be found with dashi flavouring/concentrate already added to it so that you only need to stir it into hot water, I'm not that fond of this, but it sure beats powdered miso soup. Be warned that these are usually not vegetarian. I need to get myself familiar with another kind not used in soups called moromi miso.